## Loading required package: nlme
## This is mgcv 1.8-36. For overview type 'help("mgcv-package")'.
## Loading required package: Matrix
##
## Attaching package: 'lme4'
## The following object is masked from 'package:nlme':
##
## lmList
## Package 'mclust' version 5.4.7
## Type 'citation("mclust")' for citing this R package in publications.
##
## Attaching package: 'mclust'
## The following object is masked from 'package:mgcv':
##
## mvn
## Loading required package: StanHeaders
## Loading required package: ggplot2
## rstan (Version 2.21.2, GitRev: 2e1f913d3ca3)
## For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores()).
## To avoid recompilation of unchanged Stan programs, we recommend calling
## rstan_options(auto_write = TRUE)
##
## Attaching package: 'gtools'
## The following object is masked from 'package:mgcv':
##
## scat
##
## Attaching package: 'dplyr'
## The following object is masked from 'package:nlme':
##
## collapse
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
##
## Attaching package: 'mvtnorm'
## The following object is masked from 'package:mclust':
##
## dmvnorm
##
## Attaching package: 'LaplacesDemon'
## The following objects are masked from 'package:mvtnorm':
##
## dmvt, rmvt
## The following objects are masked from 'package:gtools':
##
## ddirichlet, logit, rdirichlet
## The following objects are masked from 'package:mgcv':
##
## dmvn, rmvn
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS 10.16
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] LaplacesDemon_16.1.6 mvtnorm_1.1-2 reshape2_1.4.4
## [4] dplyr_1.0.7 gtools_3.9.2 rstan_2.21.2
## [7] ggplot2_3.3.5 StanHeaders_2.21.0-7 mclust_5.4.7
## [10] lme4_1.1-27.1 Matrix_1.3-4 mgcv_1.8-36
## [13] nlme_3.1-153 RColorBrewer_1.1-2
##
## loaded via a namespace (and not attached):
## [1] Rcpp_1.0.7 lattice_0.20-44 prettyunits_1.1.1 ps_1.6.0
## [5] assertthat_0.2.1 digest_0.6.27 utf8_1.2.2 V8_3.4.2
## [9] plyr_1.8.6 R6_2.5.1 stats4_4.0.2 evaluate_0.14
## [13] pillar_1.6.2 rlang_0.4.11 curl_4.3.2 minqa_1.2.4
## [17] rstudioapi_0.13 callr_3.7.0 nloptr_1.2.2.2 jquerylib_0.1.4
## [21] rmarkdown_2.11 splines_4.0.2 stringr_1.4.0 loo_2.4.1
## [25] munsell_0.5.0 compiler_4.0.2 xfun_0.26 pkgconfig_2.0.3
## [29] pkgbuild_1.2.0 htmltools_0.5.2 tidyselect_1.1.1 tibble_3.1.4
## [33] gridExtra_2.3 codetools_0.2-18 matrixStats_0.61.0 fansi_0.5.0
## [37] crayon_1.4.1 withr_2.4.2 MASS_7.3-54 grid_4.0.2
## [41] jsonlite_1.7.2 gtable_0.3.0 lifecycle_1.0.0 DBI_1.1.1
## [45] magrittr_2.0.1 scales_1.1.1 RcppParallel_5.1.4 cli_3.0.1
## [49] stringi_1.7.4 bslib_0.3.0 ellipsis_0.3.2 generics_0.1.0
## [53] vctrs_0.3.8 boot_1.3-28 tools_4.0.2 glue_1.4.2
## [57] purrr_0.3.4 parallel_4.0.2 processx_3.5.2 fastmap_1.1.0
## [61] yaml_2.2.1 inline_0.3.19 colorspace_2.0-2 knitr_1.34
## [65] sass_0.4.0
Load dataset
Check outliers
## The dataset contains 2649 patients with measured PfHRP2 and measured platelet counts from 4 studies
## Patients per study:
##
## Bangladesh FEAST (Uganda) Kampala (Uganda) Kilifi (Kenya)
## 172 567 492 1418
Some data cleaning
##
## FALSE TRUE
## 6 2643
##
## Bangladesh FEAST (Uganda) Kilifi (Kenya)
## 1 8 18
## A total of 27 samples have zero PfHRP2 but more than 1000 parasites per uL
## After excluding the HRP2 outliers, the dataset contains 2622 patients with measured PfHRP2 and measured platelet counts
Overview of patient characteristics
Results for Table 1 in the paper
##
## Bangladesh FEAST (Uganda) Kampala (Uganda) Kilifi (Kenya)
## 171 559 492 1400
##
## Bangladesh FEAST (Uganda) Kampala (Uganda) Kilifi (Kenya)
## 0 0 227 0 0
## 1 171 332 492 1400
## study age.lower age.median age.upper
## 1 Bangladesh 23.5 30.0 45.0
## 2 FEAST (Uganda) 1.2 2.0 3.3
## 3 Kampala (Uganda) 2.2 3.3 4.6
## 4 Kilifi (Kenya) 1.4 2.4 3.7
## study hrp2
## 1 Bangladesh 171
## 2 FEAST (Uganda) 559
## 3 Kampala (Uganda) 492
## 4 Kilifi (Kenya) 1400
## study outcome
## 1 Bangladesh 26.9
## 2 FEAST (Uganda) 11.4
## 3 Kampala (Uganda) 6.7
## 4 Kilifi (Kenya) 11.1
## study platelet.25% platelet.50% platelet.75%
## 1 Bangladesh 27.0 50.0 139.0
## 2 FEAST (Uganda) 74.5 165.0 326.0
## 3 Kampala (Uganda) 49.0 96.0 169.5
## 4 Kilifi (Kenya) 64.0 111.0 215.0
## study hrp2.25% hrp2.50% hrp2.75%
## 1 Bangladesh 1082.9050 2667.0400 6127.5550
## 2 FEAST (Uganda) 0.0000 174.7100 1952.6900
## 3 Kampala (Uganda) 588.0000 1838.4000 4097.4000
## 4 Kilifi (Kenya) 418.7393 2206.7408 5071.5299
## study wbc.25% wbc.50% wbc.75%
## 1 Bangladesh 6.900 9.000 11.000
## 2 FEAST (Uganda) 8.400 11.950 18.675
## 3 Kampala (Uganda) 7.500 10.400 15.300
## 4 Kilifi (Kenya) 8.900 12.550 19.000
## study para.25% para.50% para.75%
## 1 Bangladesh 23550 148874 348540
## 2 FEAST (Uganda) 3640 37600 153680
## 3 Kampala (Uganda) 10635 42530 198540
## 4 Kilifi (Kenya) 6099 69824 316350
##
## 1 2 3
## Bangladesh 0 0 0
## FEAST (Uganda) 466 46 21
## Kampala (Uganda) 463 4 23
## Kilifi (Kenya) 1348 41 7
## as.numeric(dat_all$platelet <= 150) hrp2.25% hrp2.50% hrp2.75%
## 1 0 24 269 1043
## 2 1 1261 3031 6035
Correlation between the platelet count and the PfHRP2 concentration
## African sites: correlation:
##
## Pearson's product-moment correlation
##
## data: log10(dat_all$platelet[ind_Africa]) and log10(dat_all$hrp2 + 1)[ind_Africa]
## t = -31.892, df = 2449, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.5690929 -0.5131183
## sample estimates:
## cor
## -0.5417058
## [1] -186.2477
## cor
## -0.54
## Bangladesh: correlation:
##
## Pearson's product-moment correlation
##
## data: log10(dat_all$platelet[!ind_Africa]) and log10(dat_all$hrp2 + 1)[!ind_Africa]
## t = -4.9404, df = 169, p-value = 1.863e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.4797402 -0.2167256
## sample estimates:
## cor
## -0.3552439
## [1] -5.72988
## cor
## -0.36
Summary plot of the biomarker data
Basic data exploration: clustering with mclust
mclust is a generic Bayesian clustering algorithm (fits multivariate normals to the data)
We merge all the data into one and fit mclust
## [1] 2477
##
## Bangladesh FEAST (Uganda) Kampala (Uganda) Kilifi (Kenya)
## 170 425 484 1398
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VEV (ellipsoidal, equal shape) model with 9 components:
##
## log-likelihood n df BIC ICL
## -6408.059 2477 73 -13386.6 -15021.76
##
## Clustering table:
## 1 2 3 4 5 6 7 8 9
## 417 27 459 196 230 433 141 305 269
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 4
## components:
##
## log-likelihood n df BIC ICL
## -3939.675 2622 23 -8060.399 -8886.706
##
## Clustering table:
## 1 2 3 4
## 776 62 182 1602
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 2
## components:
##
## log-likelihood n df BIC ICL
## -6667.391 2477 11 -13420.74 -13994.61
##
## Clustering table:
## 1 2
## 942 1535
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 2
## components:
##
## log-likelihood n df BIC ICL
## -4785.857 2477 11 -9657.677 -10326.46
##
## Clustering table:
## 1 2
## 1148 1329
##
## 1 2 3 4
## Bangladesh 24 3 0 144
## FEAST (Uganda) 147 26 179 207
## Kampala (Uganda) 149 8 0 335
## Kilifi (Kenya) 456 25 3 916
In conclusion, the distribution of the parasite count is less easily decomposed than that of the platelet count or HRP2 concentration. The HRP2 and platelet count is the only one where the estimated break is not orthogonal to either variable.
Fitting a mixture model to platelets and HRP2
Two component mixture - not including FEAST
Main model
Analysis not including the FEAST study - only severe malaria studies
We convert the platelet counts and HRP2 measurements to log10 scale. The Platelet counts then get multiplied by minus 1: this is so that increasing values correspond to more likely severe malaria. This is because the underlying stan model uses the ordered vector type to avoid label switching problems.
It was not possible to get reasonable fits from this model when we included the parasite density. This most likely results from model mis-specification. The parasite density has a lower discriminatory value for severe malaria as shown by our exploratory analyses (see github). Parasite density is mechanistically related to the HRP2 concentration. Pre-referral treatment with antimalarials (which is common in severe malaria referrals) can reduce the admission parasite density substantially (thus a much larger variance in the biomarker values for the true severe malaria subgroup). For these reasons, the main models only used the platelet counts and HRP2 concentrations.
## site_index_SMstudies
## 1 2 3
## 171 492 1400
## [1] 2063
## Priors on mean biomarker values:
## [,1] [,2]
## [1,] -2.30103 -2.00000
## [2,] 2.69897 3.39794
## Priors on standard deviations around biomarker values:
## [,1] [,2]
## [1,] 0.1 0.1
## [2,] 0.1 0.1
## Priors on prevalence of SM:
## [,1] [,2]
## [1,] 19 1
## [2,] 14 6
## [3,] 14 6
compile the stan model
Run the stan models - outputs are stored in Rout
We check convergence with the traceplots
## prevalence estimates for the three studies: model without correlation:
## [1] 93 69 64
## prevalence estimates for the three studies: model with correlation:
## [1] 94 68 59
## prevalence estimates for the three studies: model with correlation and weak priors:
## [1] 94 72 64
## prevalence estimates for the three studies: model with correlation and t-distributions:
## [1] 95 70 63
## mean values estimates for the 4 models:
## thetas_all
##
## not SM SM
## Platelet count 227 70
## PfHRP2 225 3432
## thetas_all_cor
##
## not SM SM
## Platelet count 197 71
## PfHRP2 303 3300
## thetas_all_cor_WP
##
## not SM SM
## Platelet count 214 73
## PfHRP2 211 3252
## thetas_all_cor_tdist
##
## not SM SM
## Platelet count 225 71
## PfHRP2 281 3275
## Standard deviation estimates for the 4 models:
##
## [,1] [,2]
## [1,] 0.2819436 0.7499934
## [2,] 0.3032643 0.4416046
##
## [,1] [,2]
## [1,] 0.3170179 0.8449156
## [2,] 0.3170527 0.4391073
##
## [,1] [,2]
## [1,] 0.3091195 0.7896354
## [2,] 0.3182576 0.4512951
##
## [,1] [,2]
## [1,] 0.2436313 0.7532043
## [2,] 0.2863234 0.4214951
## Correlation in Severe Malaria:
## [1] 0.3031062
## Correlation in not Severe Malaria:
## [1] 0.2345801
## [1] 1077
## [1] 448
Classifications under the bivariate normal model
Classifications under the bivariate t-distribution model
Platelet counts in not Severe Malaria and Severe Malaria
Sensitivity analysis - only African studies
Just use the two severe malaria studies from Africa
## site_index_SMAfrica
## 1 2
## 492 1400
Run model and check convergence
## 'pars' not specified. Showing first 10 parameters by default.
## [1] 65 58
## Mean estimated values across the groups:
##
## not SM SM
## Platelet count 195 73
## PfHRP2 305 3545
ROC curves for main model fit
## Platelet cutoff of 150,000 per uL has a sensitivity of 85% and a specificity of 65%
## PLasma PfHRP2 cutoff of 1000 ng/ml has a sensitivity of 88% and a specificity of 73%
Empirical ROC curves derived from patient probabilties 
ROC curves for sensitivity analyses
Three component mixture - including FEAST
This does not fit when we add parasite counts. So we are only using two biomarkers
## site_index_all_studies
## 1 2 3 4
## 171 559 492 1400
## Priors on mean biomarker values:
## [,1] [,2] [,3]
## [1,] -2.477121 -2.30103 -2.00000
## [2,] 0.000000 2.69897 3.39794
## Priors on standard deviations around biomarker values:
## [,1] [,2] [,3]
## [1,] 0.1 0.1 0.1
## [2,] 0.2 0.2 0.2
## Priors on prevalence of SM:
## [,1] [,2] [,3]
## [1,] 0.1 1 19
## [2,] 1.0 6 14
## [3,] 1.0 6 14
## [4,] 3.0 3 3
Run models:
## Estimated prevalences across the 4 studies:
##
## Bangladesh FEAST (Uganda) Kampala (Uganda) Kilifi (Kenya)
## 2.5% 89.9 30.5 65.3 59.2
## 50% 95.6 35.9 72.5 64.7
## 97.5% 98.9 41.1 79.2 69.9
##
## not SM not SM SM
## Platelet count 286 203 73
## PfHRP2 1 197 3164
## In SM the 95% pred interval for the platelet count is 17 to 308 with a mean of 73
## In SM the 95% pred interval for the PfHRP2 concentration is 414 to 24196 with a mean of 3164
## In not SM the 95% pred interval for the platelet count is 50 to 832 with a mean of 203
## In not SM the 95% pred interval for the PfHRP2 concentration is 5 to 7456 with a mean of 197
Make a P(SM) variable for subsequent analysis (continuous) and a discrete classification based on a cutoff of 0.5 
Downstream analyses
Severe malaria probability and subphenotypes
Relationship with the two main subphenotypes, cerebral malaria and severe malaria anaemia
## False diagnosis rate by coma status and by study:
## coma study SM
## 1 0 Bangladesh 0.9841270
## 2 1 Bangladesh 0.9722222
## 3 0 FEAST (Uganda) 0.3400901
## 4 1 FEAST (Uganda) 0.4403670
## 5 0 Kampala (Uganda) 0.6447368
## 6 1 Kampala (Uganda) 0.8787879
## 7 0 Kilifi (Kenya) 0.7353952
## 8 1 Kilifi (Kenya) 0.6502525
## True diagnosis rate by severe anaemia status (<15% haematocrit) and by study:
## SMA study SM
## 1 0 Bangladesh 0.9753086
## 2 1 Bangladesh 1.0000000
## 3 0 FEAST (Uganda) 0.3454106
## 4 1 FEAST (Uganda) 0.4125874
## 5 0 Kampala (Uganda) 0.8644068
## 6 1 Kampala (Uganda) 0.6835938
## 7 0 Kilifi (Kenya) 0.6469003
## 8 1 Kilifi (Kenya) 0.8596491
Main model fits - three component model
## , , study = Bangladesh
##
## class
## sickle 0 1
## 1 0 0
## 2 0 0
## 3 0 0
##
## , , study = FEAST (Uganda)
##
## class
## sickle 0 1
## 1 281 185
## 2 40 6
## 3 20 1
##
## , , study = Kampala (Uganda)
##
## class
## sickle 0 1
## 1 95 368
## 2 1 3
## 3 17 6
##
## , , study = Kilifi (Kenya)
##
## class
## sickle 0 1
## 1 399 949
## 2 26 15
## 3 6 1
Parasite densities
## [1] "FEAST (Uganda)"
## 1
## 15.42788
## [1] "Kampala (Uganda)"
## 1
## 11.49091
## [1] "Kilifi (Kenya)"
## 1
## 13.7813
## [1] "Bangladesh"
## 1
## 0.4286614
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.81224652 0.09267166 41.1371359 1.441362e-277
## P_SM 1.10526619 0.05676724 19.4701431 2.892819e-78
## studyFEAST (Uganda) -0.22016532 0.10001509 -2.2013209 2.781257e-02
## studyKampala (Uganda) -0.09041491 0.08812500 -1.0259849 3.050069e-01
## studyKilifi (Kenya) 0.07153498 0.08133515 0.8795088 3.792176e-01
## The expected fold increase in parasite density between true severe malaria (P_SM=1) and not severe malaria (P_SM=0) is:
## [1] 9.9 12.7 16.5
Admission haematocrits
## SM study hct
## 1 0 Bangladesh 35.00
## 2 1 Bangladesh 27.35
## 3 0 FEAST (Uganda) 30.00
## 4 1 FEAST (Uganda) 19.65
## 5 0 Kampala (Uganda) 11.60
## 6 1 Kampala (Uganda) 15.50
## 7 0 Kilifi (Kenya) 27.90
## 8 1 Kilifi (Kenya) 19.50
## SM study hct
## 1 0 Bangladesh 0.0
## 2 1 Bangladesh 4.2
## 3 0 FEAST (Uganda) 22.3
## 4 1 FEAST (Uganda) 26.7
## 5 0 Kampala (Uganda) 71.7
## 6 1 Kampala (Uganda) 46.2
## 7 0 Kilifi (Kenya) 9.2
## 8 1 Kilifi (Kenya) 24.5
## [1] 19
## [1] 31
## [1] 16.4
## [1] 11.6
## [1] 19.3
## [1] 28.6
Mortality
P(Severe malaria) and mortality
##
## Call:
## glm(formula = outcome ~ P_SM, family = "binomial", data = dat_all[ind,
## ])
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5061 -0.5061 -0.4959 -0.4708 2.1277
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.9906 0.1710 -11.639 <2e-16 ***
## P_SM -0.1640 0.3307 -0.496 0.62
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 397.79 on 558 degrees of freedom
## Residual deviance: 397.54 on 557 degrees of freedom
## AIC: 401.54
##
## Number of Fisher Scoring iterations: 4
##
## Call:
## glm(formula = outcome ~ P_SM, family = "binomial", data = dat_all[ind,
## ])
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.4682 -0.4505 -0.3993 -0.1943 3.1004
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.8814 0.9188 -5.313 1.08e-07 ***
## P_SM 2.7292 1.0106 2.701 0.00692 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 242.07 on 491 degrees of freedom
## Residual deviance: 229.90 on 490 degrees of freedom
## AIC: 233.9
##
## Number of Fisher Scoring iterations: 6
##
## Call:
## glm(formula = outcome ~ P_SM, family = "binomial", data = dat_all[ind,
## ])
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5620 -0.5086 -0.4520 -0.4446 2.1798
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.7657 0.1518 -11.629 <2e-16 ***
## P_SM -0.5136 0.2134 -2.406 0.0161 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 974.42 on 1399 degrees of freedom
## Residual deviance: 968.77 on 1398 degrees of freedom
## AIC: 972.77
##
## Number of Fisher Scoring iterations: 4
##
## Call:
## glm(formula = outcome ~ P_SM, family = "binomial", data = dat_all[ind,
## ])
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.8204 -0.8189 -0.8060 1.5830 1.7791
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -3.224 2.114 -1.525 0.127
## P_SM 2.308 2.168 1.065 0.287
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 199.14 on 170 degrees of freedom
## Residual deviance: 197.40 on 169 degrees of freedom
## AIC: 201.4
##
## Number of Fisher Scoring iterations: 5
Summary of lab values by inferred subgroup
## SM study age
## 1 0 Bangladesh 38.0
## 2 1 Bangladesh 30.0
## 3 0 FEAST (Uganda) 1.8
## 4 1 FEAST (Uganda) 2.4
## 5 0 Kampala (Uganda) 3.4
## 6 1 Kampala (Uganda) 3.2
## 7 0 Kilifi (Kenya) 2.3
## 8 1 Kilifi (Kenya) 2.4
## SM study log10_parasites
## 1 0 Bangladesh 100000
## 2 1 Bangladesh 79433
## 3 0 FEAST (Uganda) 251
## 4 1 FEAST (Uganda) 39811
## 5 0 Kampala (Uganda) 7943
## 6 1 Kampala (Uganda) 50119
## 7 0 Kilifi (Kenya) 10000
## 8 1 Kilifi (Kenya) 79433
## SM study log10_wbc
## 1 0 Bangladesh 8
## 2 1 Bangladesh 8
## 3 0 FEAST (Uganda) 13
## 4 1 FEAST (Uganda) 10
## 5 0 Kampala (Uganda) 13
## 6 1 Kampala (Uganda) 10
## 7 0 Kilifi (Kenya) 16
## 8 1 Kilifi (Kenya) 13
## SM study sickle.1 sickle.2 sickle.3
## 1 0 FEAST (Uganda) 281 40 20
## 2 1 FEAST (Uganda) 185 6 1
## 3 0 Kampala (Uganda) 95 1 17
## 4 1 Kampala (Uganda) 368 3 6
## 5 0 Kilifi (Kenya) 399 26 6
## 6 1 Kilifi (Kenya) 949 15 1
## SM study outcome
## 1 0 Bangladesh 0.0
## 2 1 Bangladesh 27.5
## 3 0 FEAST (Uganda) 12.0
## 4 1 FEAST (Uganda) 10.4
## 5 0 Kampala (Uganda) 2.7
## 6 1 Kampala (Uganda) 7.9
## 7 0 Kilifi (Kenya) 14.3
## 8 1 Kilifi (Kenya) 9.6
Total white blood cell counts
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.286060464 0.04589606 28.021151 2.645131e-151
## studyFEAST (Uganda) -0.124664570 0.04376642 -2.848407 4.428221e-03
## studyKampala (Uganda) -0.137776566 0.04229067 -3.257848 1.136974e-03
## studyKilifi (Kenya) -0.069168701 0.04220474 -1.638885 1.013576e-01
## SM -0.114044930 0.01120286 -10.179985 6.756193e-24
## age -0.006774506 0.00118929 -5.696263 1.361504e-08
## [1] 1.300304
## WBC > 15,000 for SM versus not SM:
##
## Family: binomial
## Link function: logit
##
## Formula:
## as.numeric(wbc > 15) ~ study + SM + s(age, k = 3)
##
## Parametric coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.02403 0.46847 4.320 1.56e-05 ***
## studyFEAST (Uganda) -2.75440 0.51248 -5.375 7.68e-08 ***
## studyKampala (Uganda) -2.71246 0.48982 -5.538 3.07e-08 ***
## studyKilifi (Kenya) -2.36279 0.49738 -4.751 2.03e-06 ***
## SM -0.71074 0.09796 -7.256 3.99e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df Chi.sq p-value
## s(age) 1.976 1.999 70.62 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.0775 Deviance explained = 6.37%
## UBRE = 0.18848 Scale est. = 1 n = 2393
## Odds-ratio:
## [1] 0.45 0.49 0.54
##
## FALSE TRUE
## Bangladesh 149 22
## FEAST (Uganda) 354 204
## Kampala (Uganda) 365 126
## Kilifi (Kenya) 870 530
Bacteraemia
Relationship between the blood culture data and the probability of severe malaria. We also look at whether WBC values predict positive blood culture
## Blood culture positive numbers by study:
## , , = Bangladesh
##
##
## 0 1
## 0 0 0
## 1 0 0
##
## , , = FEAST (Uganda)
##
##
## 0 1
## 0 273 39
## 1 165 18
##
## , , = Kampala (Uganda)
##
##
## 0 1
## 0 0 0
## 1 0 0
##
## , , = Kilifi (Kenya)
##
##
## 0 1
## 0 406 29
## 1 943 22
## Blood culture positive numbers in FEAST (all):
##
## 0 1 <NA>
## 438 57 64
## Blood culture positive numbers in FEAST (malaria RDT+):
##
## 0 1 <NA>
## 263 35 34
## Blood culture positive numbers in Kenya:
##
## 0 1
## 1349 51
## [1] 0.8855098
## main model for bacteraemia:
##
## Call:
## glm(formula = BS ~ SM + study, family = "binomial", data = dat_all[ind_mal,
## ])
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.6240 -0.3544 -0.2269 -0.2269 2.7102
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.5375 0.2074 -7.415 1.22e-13 ***
## SM -0.9108 0.2257 -4.036 5.44e-05 ***
## studyKilifi (Kenya) -1.1986 0.2318 -5.171 2.32e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 680.62 on 1697 degrees of freedom
## Residual deviance: 637.41 on 1695 degrees of freedom
## (697 observations deleted due to missingness)
## AIC: 643.41
##
## Number of Fisher Scoring iterations: 6
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.6945957 0.2564364 -6.608249 3.888921e-11
## SM -0.5781364 0.3614468 -1.599506 1.097081e-01
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.639057 0.1922129 -13.729863 6.726034e-43
## SM -1.118966 0.2888922 -3.873301 1.073710e-04
## Odds ratio for blood culture+ in SM versus not SM:
## [1] 0.26 0.40 0.63
##
## Call:
## glm(formula = BS ~ log10_wbc + study, family = "binomial", data = dat_all[ind_mal,
## ])
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5252 -0.2777 -0.2727 -0.2696 2.6240
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.1515 0.4822 -4.462 8.11e-06 ***
## log10_wbc 0.1247 0.4127 0.302 0.762
## studyKilifi (Kenya) -1.2638 0.2303 -5.488 4.08e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 680.62 on 1697 degrees of freedom
## Residual deviance: 653.53 on 1695 degrees of freedom
## (697 observations deleted due to missingness)
## AIC: 659.53
##
## Number of Fisher Scoring iterations: 6
Validation using sickle trait
##
## Call:
## glm(formula = SM ~ sickle + study, family = "binomial", data = data.frame(SM = dat_all$SM[ind],
## sickle = as.numeric(dat_all$sickle == 2)[ind], study = as.factor(dat_all$study)[ind]))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7857 -1.0038 0.6737 0.8384 1.9731
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.42308 0.09322 -4.539 5.66e-06 ***
## sickle -1.36948 0.25277 -5.418 6.03e-08 ***
## studyKampala (Uganda) 1.79061 0.14762 12.130 < 2e-16 ***
## studyKilifi (Kenya) 1.28798 0.10953 11.759 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 3082.3 on 2367 degrees of freedom
## Residual deviance: 2828.8 on 2364 degrees of freedom
## AIC: 2836.8
##
## Number of Fisher Scoring iterations: 4
## Odds ratio for SM for HbAS versus HbAA:
## [1] 0.15 0.25 0.42
## [1] "Kilifi (Kenya)"
## [1] 0.13 0.24 0.46
## [1] "Kampala (Uganda)"
## [1] 0.08 0.77 7.53
## [1] "FEAST (Uganda)"
## [1] 0.09 0.23 0.55
## , , = Bangladesh
##
##
## 1 2 3
## 0 0 0 0
## 1 0 0 0
##
## , , = FEAST (Uganda)
##
##
## 1 2 3
## 0 281 40 20
## 1 185 6 1
##
## , , = Kampala (Uganda)
##
##
## 1 2 3
## 0 95 1 17
## 1 368 3 6
##
## , , = Kilifi (Kenya)
##
##
## 1 2 3
## 0 399 26 6
## 1 949 15 1
##
## Call:
## glm(formula = SM ~ sickle + study, family = "binomial", data = data.frame(SM = dat_all$SM[ind],
## sickle = as.numeric(dat_all$sickle == 3)[ind], study = as.factor(dat_all$study)[ind]))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7815 -1.0054 0.6765 0.8380 2.4244
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.41900 0.09432 -4.442 8.90e-06 ***
## sickle -2.46542 0.40554 -6.079 1.21e-09 ***
## studyKampala (Uganda) 1.77702 0.14754 12.045 < 2e-16 ***
## studyKilifi (Kenya) 1.28482 0.11151 11.522 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 3018.5 on 2327 degrees of freedom
## Residual deviance: 2773.9 on 2324 degrees of freedom
## AIC: 2781.9
##
## Number of Fisher Scoring iterations: 4
## Odds ratio for SM for HbSS versus HbAA:
## [1] 0.04 0.08 0.19
## [1] "Kilifi (Kenya)"
## [1] 0.01 0.07 0.58
## [1] "Kampala (Uganda)"
## [1] 0.03 0.09 0.24
## [1] "FEAST (Uganda)"
## [1] 0.01 0.08 0.57
##
## Call:
## glm(formula = SM ~ sickle + study, family = "binomial", data = data.frame(SM = dat_all$SM,
## sickle = dat_all$sickle, study = as.factor(dat_all$study)))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7894 -1.0023 0.6713 0.8394 2.4717
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.8628 0.1951 4.423 9.72e-06 ***
## sickle -1.2897 0.1613 -7.994 1.31e-15 ***
## studyKampala (Uganda) 1.8025 0.1452 12.416 < 2e-16 ***
## studyKilifi (Kenya) 1.2889 0.1091 11.816 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 3177.2 on 2418 degrees of freedom
## Residual deviance: 2869.2 on 2415 degrees of freedom
## (203 observations deleted due to missingness)
## AIC: 2877.2
##
## Number of Fisher Scoring iterations: 4
## Odds ratio for SM for additional S allele (additive model):
## [1] 0.20 0.28 0.38
Compare with previously published probabilities
##
## Pearson's product-moment correlation
##
## data: model1_probs$P_SM1 and dat_Kilifi$P_SM
## t = 31.253, df = 1398, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.6094134 0.6711674
## sample estimates:
## cor
## 0.6413278
Save output
Optimal combination of thresholds
Plot results
## [1] 5.18 40.44
## [1] 50.11 92.21
## [1] 77 94
## There are 0 combinations that have a true positive rate greater than 75 and a false positive rate less than 10
## plt_thresh hrp2_thresh x1_trans x2_trans TP FP ProbSM
## 18231 150 1000 -2.176091 3 75.96 13.43 89
Predicting Severe Malaria using HRP2, parasite counts and HCT
Slightly random: predict SM status using HRP2 and HCT
##
## Family: binomial
## Link function: logit
##
## Formula:
## SM ~ s(hct, k = 3) + s(log10(hrp2 + 1), k = 3) + s(log10_parasites,
## k = 3) + s(study, bs = "re")
##
## Parametric coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.010 6.051 -0.167 0.867
##
## Approximate significance of smooth terms:
## edf Ref.df Chi.sq p-value
## s(hct) 1.949 1.997 16.70 0.000224 ***
## s(log10(hrp2 + 1)) 1.978 1.999 290.23 < 2e-16 ***
## s(log10_parasites) 1.747 1.936 16.64 0.000142 ***
## s(study) 2.979 3.000 47.87 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.864 Deviance explained = 83.1%
## UBRE = -0.77747 Scale est. = 1 n = 2474